### Nuprl Lemma : implies_l_member_append

`∀T:Type. ∀l1,l2:T List. ∀v:T.  (((v ∈ l1) ∨ (v ∈ l2)) `` (v ∈ l1 @ l2))`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` append: `as @ bs` list: `T List` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` l_member: `(x ∈ l)` or: `P ∨ Q` exists: `∃x:A. B[x]` cand: `A c∧ B` member: `t ∈ T` uall: `∀[x:A]. B[x]` prop: `ℙ` top: `Top` ge: `i ≥ j ` nat: `ℕ` decidable: `Dec(P)` false: `False` le: `A ≤ B` and: `P ∧ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` squash: `↓T` int_seg: `{i..j-}` lelt: `i ≤ j < k` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` less_than: `a < b` subtract: `n - m`
Lemmas referenced :  l_member_wf list_wf istype-universe length-append istype-void non_neg_length nat_properties decidable__lt length_wf full-omega-unsat intformand_wf intformnot_wf intformless_wf itermVar_wf itermAdd_wf intformle_wf itermConstant_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_wf equal_wf squash_wf true_wf select_append_front decidable__le le_wf istype-less_than subtype_rel_self iff_weakening_equal append_wf select_wf add_nat_wf length_wf_nat add-is-int-iff set_subtype_base int_subtype_base intformeq_wf int_formula_prop_eq_lemma false_wf select_append_back add-associates minus-one-mul add-swap add-mul-special zero-mul add-zero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  sqequalHypSubstitution unionElimination thin productElimination sqequalRule Error :unionIsType,  Error :universeIsType,  cut introduction extract_by_obid isectElimination hypothesisEquality hypothesis Error :inhabitedIsType,  instantiate universeEquality Error :dependent_pairFormation_alt,  Error :isect_memberEquality_alt,  voidElimination because_Cache setElimination rename dependent_functionElimination addEquality natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :lambdaEquality_alt,  int_eqEquality independent_pairFormation applyEquality imageElimination equalityTransitivity equalitySymmetry Error :dependent_set_memberEquality_alt,  Error :productIsType,  imageMemberEquality baseClosed Error :equalityIsType1,  applyLambdaEquality pointwiseFunctionality promote_hyp intEquality baseApply closedConclusion

Latex:
\mforall{}T:Type.  \mforall{}l1,l2:T  List.  \mforall{}v:T.    (((v  \mmember{}  l1)  \mvee{}  (v  \mmember{}  l2))  {}\mRightarrow{}  (v  \mmember{}  l1  @  l2))

Date html generated: 2019_06_20-PM-02_57_09
Last ObjectModification: 2018_10_17-AM-10_43_40

Theory : continuity

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